.. Objectives
.. ----------

.. By the end of this tutorial you will --

.. 1. Get an idea of the range of things for which Sage can be used. 
.. #. Know some of the functions for Calculus
.. #. Get some insight into Graphs in Sage. 


.. Prerequisites
.. -------------

.. Getting Started -- Sage  
     
Script
------

.. L1

{{{ Show the title slide }}}

.. R1

Hello Friends and  Welcome to the tutorial on 'Using Sage for Calculus'.

.. L2

{{{ show the 'objectives' slide }}} 

.. R2

At the end of this tutorial, you will be able to,

 1. Learn the range of things for which Sage can be used. 
 #. Perform integrations & other Calculus in Sage.
 #. Perform matrix algebra in sage.
 
.. L3

{{{ show the 'pre-requisite' slide }}}

.. R3

Before beginning this tutorial,we would suggest you to complete the 
tutorial on "Getting started with Sage".  

Let us begin with Calculus. We shall be looking at limits,
differentiation, integration, and Taylor polynomial.

.. L4

{{{ open sage notebook }}}

.. R4

We have our Sage notebook running. In case, you don't have it running,
start is using the command, ``sage --notebook``.

.. R5

To begin with, let us find the limit of the function x*sin(1/x), at x=0.
To do this we say

.. L5
::

    lim(x*sin(1/x), x=0)

.. R6

As expected, we get the limit to be 0. 

It is also possible to limit a point from one direction. For
example, let us find the limit of 1/x at x=0, when approaching from
the positive side.

.. L6
::

    lim(1/x, x=0, dir='right')

.. R7

We get the limit from positive side.
To find the limit from the negative side, we say,

.. L7
::

    lim(1/x, x=0, dir='left')   

.. L8

{{{ Show the 'differential expression' slide }}}

.. R8

Let us now see how to perform differentiation, using Sage. We shall 
find the differential of the expression ``exp(sin(x^2))/x`` w.r.t ``x``.
For this, we shall first define the expression, and then use the ``diff`` 
function to obtain the differential of the expression. So, switch to the sage
notebook and type

.. L9
::

    var('x')
    f = exp(sin(x^2))/x
    diff(f, x)

.. R9

And we get the expected differential of the expression.

.. L10

{{{ Show the slide 'Partial Differentiation' }}}

.. R10

We can also obtain the partial differentiation of an expression with one of the
vriables. Let us differentiate the expression
``exp(sin(y - x^2))/x`` w.r.t x and y. Switch to sage notebook and type

.. L11
::

    var('x y')
    f = exp(sin(y - x^2))/x
    diff(f, x)
    diff(f, y)

.. R11

Thus we get our partial differential solution.

.. L12

{{{ Show the 'integration' slide }}}

.. R12

Now, let us look at integration. We shall use the expression obtained
from the differentiation that we calculated before, ``diff(f, y)``
which gave us the expression ---``cos(-x^2 + y)*e^(sin(-x^2 + y))/x``. 
The ``integrate`` command is used to obtain the integral of an 
expression or function. So, switch to sage notebook and type.

.. L13
{{{ Switch to sage }}}
::

    integrate(cos(-x^2 + y)*e^(sin(-x^2 + y))/x, y)

.. R13

As we can see, we get back the correct expression. The minus sign being 
inside or outside the ``sin`` function doesn't change much. 

Now, let us find the value of the integral between the limits 0 and
pi/2. 

.. L14
::

    integral(e^(sin(-x^2 + y))*cos(-x^2 + y)/x, y, 0, pi/2)

.. R14

Hence we get our solution for the definite integration.
Let us now see how to obtain the Taylor expansion of an expression
using sage. We will obtain the Taylor expansion of ``(x + 1)^n`` up to
degree 4 about 0.

.. L15
::

    var('x n')
    taylor((x+1)^n, x, 0, 4)

.. R15

We easily got the Taylor expansion,using the function ``taylor()``.
This brings us to the end of the features of Sage for Calculus, that
we will be looking at. 

.. L16

{{{ Show the 'More on Calculus' slide }}}

.. R16

For more on calculus you may look at the Calculus quick-ref from the Sage
documentation at the given link.

.. L17

{{{ show the 'Equation' slide }}}

.. R17

Next let us move on to Matrix Algebra. 
Let us begin with solving the equation ``Ax = v``, where A is the
matrix ``matrix([[1,2],[3,4]])`` and v is the vector
``vector([1,2])``. 

.. R18

To solve the equation, ``Ax = v`` we simply say

.. L18

{{{ Switch back to sage notebook page }}}
::

    A = matrix([[1,2],
                [3,4]])
    v = vector([1,2])
    x = A.solve_right(v)
    x

.. R19

To solve the equation, ``xA = v`` we simply say.
The left and right here, denote the position of ``A``, relative to x. 

.. L19
::

    x = A.solve_left(v)
    x

.. L20

{{{ show the 'Summary' slide }}}

.. R20

This brings us to the end of this tutorial. In this tutorial we have learned to

1. Use functions like lim(), integrate(), integral(), solve()
#. Use sage for performing matrix algebra, integrations & other calculus 
operations using the above mentioned functions.

.. L21

{{{ Show the 'Evaluation' slide }}}

.. R21

Here are some self assessment questions for you to solve.

 1. How do you find the limit of the function x/sin(x) as x tends to 0 from the
    negative side.

 #. Solve the system of linear equations
    x-2y+3z = 7
    2x+3y-z = 5
    x+2y+4z = 9
   
Try the xercises and switch to next slide for solutions.
    
.. L22

{{{ Show the 'Solutions' slide }}}

.. R22

 1. To find the limit of the function x/sin(x) as x tends to 0 from negative
side, use the lim function as: lim(x/sin(x), x=0, dir'left')

 #. A = Matrix([1, -2, 3], [2, 3, -1], [1, 2, 4]])
    b = vector([7, 5, 9])
    x = A.solve_right(b)
    x

.. L23

{{{ Show the 'FOSSEE' slide }}}

.. R23

FOSSEE is Free and Open-source Software for Science and Engineering Education.
The goal of this project is to enable all to use open source software tools.
For more details, please visit the given link.

.. L24

{{{ Show the 'About the Spoken Tutorial Project' slide }}}

.. R24

Watch the video available at the following link. It summarizes the Spoken 
Tutorial project. If you do not have good bandwidth, you can download and 
watch it.

.. L25

{{{ Show the 'Spoken Tutorial Workshops' slide }}}

.. R25

The Spoken Tutorial Project Team conducts workshops using spoken tutorials,
gives certificates to those who pass an online test.

For more details, please write to contact@spoken-tutorial.org

.. L26

{{{ Show the 'Acknowledgements' slide }}}

.. R26

Spoken Tutorial Project is a part of the "Talk to a Teacher" project.
It is supported by the National Mission on Education through ICT, MHRD, 
Government of India. More information on this mission is available at the 
given link.

.. L27

{{{Show the 'Thank you' slide }}}

.. R27

Hope you have enjoyed this tutorial and found it useful.
Thank you!